Preheat oven to 350 degrees F. Grease a 9x12-inch baking dish. Heat the olive oil in a large skillet over medium heat, and cook the onion and garlic until the onion is translucent, about 5 minutes. Stir in the zucchini, yellow squash, and red bell pepper. Cook, stirring frequently, until the zucchini and squash are tender, about 10 minutes. Stir in the black beans, green salsa, cumin, and cayenne pepper. Cook and stir until the filling thickens, 5 to 8 more minutes.
Arrange tortillas on a work surface. Divide one package cheese among tortillas. Spoon filling into each tortilla, over the cheese, in a line down the center. Roll up the tortillas, and place them into the prepared baking dish with the seam sides down. (Roll them tightly. It will be a snug fit.)
Bake in the preheated oven until the cheese is melted and the tortillas are heated through, about 15 minutes. Sprinkle the other package of shredded cheese over the tortillas before serving. (From our kitchen: We baked the burritos 10 minutes, then topped with cheese and baked another 5 minutes so that the cheese on top melted.)
Option: make the filling ahead so you can just fill and bake!